A copper rod of $88\; \mathrm{cm}$ and an aluminum rod of unknown length have their increase in length independent of increase in temperature. The length of aluminum rod is….$cm$

$( \alpha_{Cu}=1.7 \times 10^{-5}\; \mathrm{K}^{-1}$ and $\alpha_{Al}=2.2 \times 10^{-5} \;\mathrm{K}^{-1} ) $

At $40\,^oC$, a brass wire of $1\, mm$ is hung from the ceiling. A small mass, $M$ is hung from the free end of the wire. When the wire is cooled down from $40\,^oC$ to $20\,^oC$ it regains its original length of $0.2\, m$. The value of $M$ is close to ……..$kg$ (Coefficient of linear expansion and Young's modulus of brass are $10^{-5}/^oC$ and $10^{11}\, N/m^2$, respectively; $g = 10\, ms^{-2}$)

A surveyor's $30$-$m$ steel tape is correct at some temperutre. On a hot day the tape has expanded to $30.01$ $m$. On that day, the tape indicates a distance of $15.52$ $m$ between two points. The true distance between these points is :-

A flask of volume $10^3 cc$ is completely filled with mercury at $0 ^oC$. The coefficient of cubical expansion of mercury is $180 × 10^{-6} / ^oC$ and that of glass is $40 × 10^{-6}/ ^oC$. If the flask is now placed in boiling water at $100\ ^oC$, …….. $cc$ mercury will overflow 

Ratio among linear expansion coefficient ($\alpha$), areal expansion coefficient ($\beta$) and volume expansion coefficient ($\gamma$) is